Constructive Method for Polynomial Extensions in Two Dimensions

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ProcediaProcedia Engineering 00 (2011) 000–000 Engineering 31 (2012) 366 – 372

Procedia Engineering www.elsevier.com/locate/procedia

International Conference on Advances in Computational Modeling and Simulation

Constructive Method for Polynomial Extensions in Two Dimensions Jianming zhang* Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming, 650500, China

Abstract Polynomial extensions play a vital role in numerical analysis of some high-order approximation, such as the p version and the h-p version of the finite element method and spectral methods. In this paper, we construct explicitly polynomial extensions on a triangle T and a square S , which lift a polynomial defined on a side  or on whole boundary of T or S . These extension operators from H 001/2 () to H 1 (T ) or H 1 (S ) and from H 1/2 (T ) to H 1 (T ) or from H 1/2 (S ) to H 1 (S ) are continuous.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Kunming University of Science and Technology Keywords: finite element method, polynomial extension, continuous operator,convolution, compatible. Sobolev spaces

Nomenclature Integer

p

p0

C

A constant independent of f and p

Pp1 ()

The sets of polynomials of total degree  p on domain 

Pp2 ()

The sets of polynomials of separate degree  p on domain 

Pp (i )

The polynomial space of degree  p over

i

* Corresponding author. Tel.: +86-15824158919; fax: +86-871-330 3561. E-mail address: [email protected].

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.01.1038

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The side of the triangle T ,1  i  3

i

1. Introduction In analysis of the high-order finite element method (FEM), such as the p and h-p versions of FEM and the spectral element method, we need to construct a globally continuous and piecewise polynomial which has the optimal estimation for its approximation error and satisfies homogeneous or non-homogeneous Dirichlet boundary conditions. The construction of such a polynomial is started with local polynomial projections on each element for the best approximation. A simple union of local polynomials projections is not globally continuous and does not satisfy the Dirichlet boundary conditions. For the continuous Galerkin method in two and three dimensions, we have to adjust these local polynomial projections by a special technique called polynomial extension or lifting. Hence, it is essential for us to build a polynomial extension compatible to FEM subspaces, by which the union of local polynomial projections can be modified to a globally continuous polynomial without degrading the best order of approximation error. The polynomial extensions together with local projections have led to the best estimation in the approximation error for the p FEM [3, 4, 5, 6,10] and h-p FEM and the best condition number for the preconditioning of the p FEM [4] and the h-p FEM [11]. Babuška and Suri [6] proposed an extension F on an equilateral triangle T = {(x, y) | - y ≤ x ≤ y , 3

0≤ y≤

3 2

3

} with I = (0, 1) as one of its sides, F [ f ] ( x, y) =

3 2y



=  f (t ) H ( x  t , y)dt = ( f  H (, y))( x)

with the characteristic function H ( x, y)  3 for  y  x  y , and H ( x, y)  0 , otherwise. This 2y

extension realizes a continuous mapping F [ f ] |I

3

3

H 1/2 ( I )  H 1 (T ) such

that F [ f ]  Pp1 (T ) for f  Pp ( I ) and

= f. Using this extension operator of convolution type they were able to prove implicitly the

existence of the continuous extension operator R : H001/2 ( I )  H 1 (T ) [2, 6] such that R[ f ]  Pp1 (T ) for f  Pp0 ( I )  { Pp ( I ) |  (0)   (1)  0} ,

and  R[ f ] |I f , R[ f ] |T \ I 0 . Incorporating the operator R on the triangle T

and a bilinear mapping of a standard square S = (−1, 1)2 onto a truncated triangle T (a trapezoid), they generalized the polynomial extension R to the square S , which realizes a continuous mapping 1/2  R[ f ] |I f , R[ f ] |S \ I 0 . H00 ()  H 1 (S ) such that R[ f ]  Pp2 (S ) for f  Pp0 ( I ) , and In

1991

Babuška

et al.[2] proved implicitly the existence of continuous operator or S based on the polynomial extension F of the convolution type such that E[ f ]  Pp2 () for f  Pp () , and E[ f ] |  f . This polynomial extension from the whole boundary to the E : H 1/2 ()  H 1 (),   T

interior of the standard domain was utilized for preconditioning of the p FEM [2]. The proof of the continuity of the operator F in [2, 6] is direct and straightforward by the Fourier Transform because the operator F is defined explicitly. On the contrary the proof for the operators R and E are not straightforward and constructive since they are not explicitly constructed. Recently we have noted that the polynomial extension of convolution type has been successfully generalized to tetrahedrons [24] in three dimensions. Muňoz-Sola creatively developed the polynomial extension of convolution type on a tetrahedron by constructing explicitly the extension operator R, for which the proof for the continuity

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is explicit and constructive. Although the polynomial extension on a tetrahedron can not be generalized to construct extensions on prisms and hexahedrons, we found that the structure of the extension operator R introduced there can be adopted for the polynomial extension R on a triangle in two dimensions, which can make the proof constructive and straightforward. In this paper we would explicitly construct the operators R and E on a triangle T and a square S. 2. Polynomial extension on a triangle 2.1. Polynomial extension R from one side of a triangle  T {( x, y) | 0  y  1  x,0  x  1} be a right triangle in  2 , Let 1

1

f  L2 (1 )  {x 2 (1  x) 2 f ( x)  L2 (1 )}, 1  I  [0,1],   1/ 2. F [ f ] ( x, y) 

Obviously,

We define an operator F by

1 x y f ( )d y x

(1)

F [ f ] ( x,0)  f ( x), and F [ f ] ( x, y)  Pp1 (T ) if f  Pp (1 ).

Fig.1. A standard triangular domain T

Lemma 1 Let F [ f ] is defined as in (1). Then for f  L2 (1 ) there hold 1

(2)

|| F [ f ] ( x, y) ||L2 (T )  C || x 2 f ( x) ||L2 (  ) 1

and 1

|| F [ f ] ( x, y) ||L2 (T )  C || (1  x) 2 f ( x) ||L2 (  ) 1

(3)

where C is a constant independent of f and p . Theorem 1 Let F be the operator defined as in (1). Then F is a continuous mapping: H t (1 )  H

t

1 2

(T ), t  0,

1 2

such that F [ f ] ( x, y) |  f ( x) for f ( x)  H t (1 ), 1

|| F

[f]

( x, y) ||H1 (T )  C || f ( x) ||

1

H 2 ( 1 )

,

and || F [ f ] ( x, y) ||

We further introduce an operator RT[ f ] ( x, y)  x(1  x  y) F

[

f ]  (1 )

1

H 2 (T )

RT

( x, y) 

 C || f ( x) ||L2 ( ) . 1

by x(1  x  y) x y f ( ) x  (1   )d . y

(4)

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For all f  H 002 . Obviously, RT[ f ] ( x, y) |  f ( x) and RT[ f ] ( x, y) |   0. 2

1

3

1 2 00

Lemma 2 Let f ( x)  H (1 ), and let RT[ f ] ( x, y) be given as in (4). Then 1

(5)

|| RT[ f ] ( x, y) ||L2 (T )  C || x 2 f ( x) ||L2 (  ) 1

and 1

(6)

|| RT[ f ] ( x, y) ||L2 (T )  C || (1  x) 2 f ( x) ||L2 (  ) 1

1

Lemma 3 Let f ( x)  H 002 (1 ), and let RT[ f ] ( x, y) be given as in (4). Then ||

RT[ f ] ( x, y) ||L2 (T )  C || f ( x) || 1 . 2 ( ) x H 00 1

(7)

Lemma 4 Let f ( x)  Pp (1 ) vanishing at the endpoints of ||

1 ,

and let RT[ f ] ( x, y) be given by (4). Then

RT[ f ] || 2  C || f || 1 . 2 ( ) y L (T ) H 00 1

(8)

Theorem 1 Let f ( x)  Pp0 (1 ) , and let RT[ f ] ( x, y) be constructed as in (4). Then RT[ f ] ( x, y)  Pp1,0 (T )  {  Pp1 (T ) |  |2 3  0}  Pp1 (T )

|| RT[ f ] ( x, y) ||H1 (T )  C || f ||

2.2. Polynomial extension

ET

and RT[ f ] ( x, y) | 1 f ( x) , and (9)

.

1 2 ( ) H 00 1

from whole boundary of a triangle

We shall construct an extension

of a polynomial defined on whole boundary T of a triangle T ,

ET 1 2

and the ET is a continuous operator: H (T )  H 1 (T ) ,which lifts a polynomial f  Pp (T )  { f  C 0 (T ),1  i  3} to the interior of T . 1

We introduce the Sobolev space H 002 (1 , 0) by 1

1



1

H 002 (1 , 0) {v  H 2 (1 ) | x 2 v  L2 (1 )}

(10)

with the norm || v  ||2 1 || v ||2 1 2 (  ,0)) H 00 1

H 2 ( 1 )



1

| v( x) |2 dx . x

1 2 00

x x y f ( ) d . Then Lemma 5 Let f ( x)  H (1, 0) , and let R1f ( x, y)  x



y

[f] 1

|| R ( x, y) ||H1 (T )  C || f ( x) ||

1 2 (  ,0) H 00 1

.

(11)

1

Theorem 2 There exists a linear operator ET : H 2 (T )  H 1 (T ) such that ET[ f ] |T  f and ET[ f ]  Pp (T ) for f  Pp (T ) ,

and || ET[ f ] ||H1 (T )  C || f ||

1

H 2 ( T )

(12)

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3. Polynomial extension on a square 3.1. Polynomial extension

from one side of a square

RS

We shall construct a polynomial extension RS on a square S  (1,1)2 , which maps a polynomial defined on one side of S to the interior of S , and this extension operator is

Fig.2 Mapping of square S onto

Let TH with 0  H  1 be a trapezoid with edge  1, 2 ,  3 and  4 as shown in Fig.2.where  2 and  4 are portion of the side 2 and 3 of the right triangle T , respectively. A bilinear mapping M : maps S onto

TH

and the sides

For f  Pp (1 ) , let

i

onto the sides  i ,1  i  4 . we define

f ( x ) f (2 x  1)  f  M 1 , 

FS[ f ]  F [ f ]  M

.

(13)

Then we have following lemma. 1

Lemma 6 Let f  H 2 (1 ) , and let FS[ f ] be defined in (13). Then FS[ f ] ( x, y)  Pp2 (S ), FS[ f ] | 1 f and || FS[ f ] ||H t ( S )  C || f ||

H

t

For f  Pp (1 ) vanishing at the endpoints of

1 2 ( ) 1

1 ,

let

] RS[ f  ( x, y) U ( x, y )  U ( x,1)

Lemma 7 Let f  Pp0 ( 1 ) , and let [ f ]

|| R



R[ f ] ( x, y )

||H1 (

3

 f ( x ) f (2x  1)  f  M 1,U ( x, y ) R[ f ]  M

2 p

R ( x, y)  P (S ), R

[f] S

|| R

(15)

be given as in (4). Then

 C || f || )

|  f and vanishes

[f] 1 S

,we introduce

y 1 2

1 2 ( ) H 00 1

Theorem 3 Let f  Pp (1 ) vanishes at the endpoints of [f] S

(14)

,t  0,1

1 and RS[ f ] be

on other sides of S , and

||H1 ( S )  C || f ||

1 2 ( ) H 00 1

defined in (15). Then

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3.2. Polynomial extension

ES

from whole boundary of a square 1

We shall next to discuss the existence of a continuous extension ES : H 2 (S )  H 1 (S ) which lifts f  Pp (S ) {  C 0 (S ) |  | i  fi  Pp (i ),1  i  4} to interior of S . By Schwarz inequality, we have the following lemma: Lemma 8 For g  L2 ( I ), I  (0,1) , there holds for 0  h  1 2

1 1 x h g (t )dt | dx  C  x | g ( x) |2 dx 0 0 h x 2 1h 1 xh 1 2 0 | h x g (t )dt | dx  C 0 (1  x) | g ( x) | dx



and

1h

|

.

Lemma 9 Let S be square with sides 1 to  4 and f  Pp (1 ) . Then there exists U  Pp2 (S ) such that U  f on the opposite side  3 and || U ||H1 ( S )  C || f ||

1

H 2 ( 1 )

Theorem 4 Let

S  [1,1]2

be the square with sides denoted by  i ,1  i  4 , and let f  Pp (S ) . Then there

exists an extension ES such that ES[ f ]  Pp2 (S ) and ES[ f ] |S  f , and || ES[ f ] ||H1 ( S )  C || f ||

1

H 2 ( S )

where C is a constant independent of f and p . 4. Conclusion In this paper, we introduced the idea of constructive method for polynomial extensions in two dimensions and reported our some results about explicit construction of polynomial extensions in two dimensions. Furthermore, these results can be applied to p and h-p versions finite element method. Because of the limitation of the paper’s length, the details of proof of theorems in previous sections will be presented in forthcoming paper [9]. References [1] M. Ainsworth and L. Demkowicz, Explicit polynomial preserving trace liftings on a triangle, London Mathematical Society,submitted. See also ICES Report 03-47. [2] I. Babuška, A. Craig, J. Mandel, and J. Pitkäranta, Efficient preconditioning for the p version finite element method in two dimensions, SIAM J. Numer. Anal., 28(1991), pp. 624-661. [3] I. Babuška, and B.Q. Guo, Direct and inverse approximation theorems of the p version of finite element method in the framework of weighted Besov spaces, Part 1: Approximability of functions in weighted Besov spaces, SIAM J. Numer. Anal, 39 (2002), pp. 1512-1538. [4] I. Babuška, and B.Q. Guo, Direct and inverse approximation theorems of the p-version of the finite element method in the framework of weighted Besov spaces, Part 2: Optimal convergence of the p-version of the finite element method, Math. Mod. Meth. Appl. 12 (2002), pp. 689-719 [5] I. Babuška and M. Suri, The optimal convergence rate of the p-version of the finite element method, SIAM J. Numer Anal. 24 (1991), pp. 750-776.

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[6] I. Babuška and M. Suri, The h-p version of the finite element method with quasiuniform meshes, RAIRO Mod´el. Math. Anal. Num´er., 21(1987), pp. 199-238. [7] I. Babuška, M. Szab´o and N. Katz, The p-version of the finite element method, SIAM J. Numer Anal. 18 (1981), pp. 515545. [8] C. Bernardi and Y. Maday, Spectral Methods, Handbook of Numerical Analysis. Vol V, Part 2, Eds. Ciarlet, P.G. and Lion, J.L., 1997, pp. 209-475. [9] B.Q. Guo and J. Zhang, Constructive proof for polynomial extensions in two dimensions, preprint. [10] R. Muñoz-Sola, Polynomial liftings on a tetrahedron and applications to the h-p version of the finite element method in three dimensions , SIAM J. Numer. Anal. 34 (1997), pp. 282-314. [11] B.Q. Guo, and W. Cao, A preconditioner for the h-p version of the finite element method in two dimensions, Numer. Math., 75(1996), pp. 59-77.